<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 4.2.0">
  <link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png">
  <link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-next.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-next.png">
  <link rel="mask-icon" href="/images/logo.svg" color="#222">

<link rel="stylesheet" href="/css/main.css">


<link rel="stylesheet" href="/lib/font-awesome/css/font-awesome.min.css">


<script id="hexo-configurations">
  var NexT = window.NexT || {};
  var CONFIG = {
    hostname: new URL('http://yoursite.com').hostname,
    root: '/',
    scheme: 'Pisces',
    version: '7.6.0',
    exturl: false,
    sidebar: {"position":"left","display":"post","padding":18,"offset":12,"onmobile":false},
    copycode: {"enable":false,"show_result":false,"style":null},
    back2top: {"enable":true,"sidebar":false,"scrollpercent":false},
    bookmark: {"enable":false,"color":"#222","save":"auto"},
    fancybox: false,
    mediumzoom: false,
    lazyload: false,
    pangu: false,
    comments: {"style":"tabs","active":null,"storage":true,"lazyload":false,"nav":null},
    algolia: {
      appID: '',
      apiKey: '',
      indexName: '',
      hits: {"per_page":10},
      labels: {"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}
    },
    localsearch: {"enable":false,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},
    path: '',
    motion: {"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}}
  };
</script>

  <meta name="description" content="Overview　　SRCNN是深度用在超分辨率重建上的开山之作。其指出传统的基于稀疏矩阵的方法可以看做一个深层神经网络。SRCNN是一个轻量级的网络，只有三个卷积神经网络(CNN)。作者在训练时将图片转化为YCrCb三个通道，只对Y通道进行训练。因为人眼对Y通道比较敏感。">
<meta property="og:type" content="article">
<meta property="og:title" content="SRCNN">
<meta property="og:url" content="http://yoursite.com/2019/12/24/SRCNN/index.html">
<meta property="og:site_name" content="SRCNN">
<meta property="og:description" content="Overview　　SRCNN是深度用在超分辨率重建上的开山之作。其指出传统的基于稀疏矩阵的方法可以看做一个深层神经网络。SRCNN是一个轻量级的网络，只有三个卷积神经网络(CNN)。作者在训练时将图片转化为YCrCb三个通道，只对Y通道进行训练。因为人眼对Y通道比较敏感。">
<meta property="og:locale" content="en_US">
<meta property="og:image" content="http://yoursite.com/images/SRCNN_model.png">
<meta property="article:published_time" content="2019-12-24T08:54:39.000Z">
<meta property="article:modified_time" content="2019-12-30T07:32:14.495Z">
<meta property="article:author" content="Z.J. Jiang">
<meta property="article:tag" content="SISR">
<meta property="article:tag" content=" FH">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="http://yoursite.com/images/SRCNN_model.png">

<link rel="canonical" href="http://yoursite.com/2019/12/24/SRCNN/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome: false,
    isPost: true
  };
</script>

  <title>SRCNN | SRCNN</title>
  






  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript>

</head>

<body itemscope itemtype="http://schema.org/WebPage">
  <div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-meta">

    <div>
      <a href="/" class="brand" rel="start">
        <span class="logo-line-before"><i></i></span>
        <span class="site-title">SRCNN</span>
        <span class="logo-line-after"><i></i></span>
      </a>
    </div>
        <p class="site-subtitle">SISR-FH</p>
  </div>

  <div class="site-nav-toggle">
    <div class="toggle" aria-label="Toggle navigation bar">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>
</div>


<nav class="site-nav">
  
  <ul id="menu" class="menu">
        <li class="menu-item menu-item-home">

    <a href="/" rel="section"><i class="fa fa-fw fa-home"></i>Home</a>

  </li>
        <li class="menu-item menu-item-tags">

    <a href="/tags/" rel="section"><i class="fa fa-fw fa-tags"></i>Tags</a>

  </li>
        <li class="menu-item menu-item-categories">

    <a href="/categories/" rel="section"><i class="fa fa-fw fa-th"></i>Categories</a>

  </li>
        <li class="menu-item menu-item-archives">

    <a href="/archives/" rel="section"><i class="fa fa-fw fa-archive"></i>Archives</a>

  </li>
  </ul>

</nav>
</div>
    </header>

    
  <div class="back-to-top">
    <i class="fa fa-arrow-up"></i>
    <span>0%</span>
  </div>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content">
            

  <div class="posts-expand">
      
  
  
  <article itemscope itemtype="http://schema.org/Article" class="post-block " lang="en">
    <link itemprop="mainEntityOfPage" href="http://yoursite.com/2019/12/24/SRCNN/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/images/avatar.gif">
      <meta itemprop="name" content="Z.J. Jiang">
      <meta itemprop="description" content="about the single image super-resolution and face hallucination">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="SRCNN">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          SRCNN
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="fa fa-calendar-o"></i>
              </span>
              <span class="post-meta-item-text">Posted on</span>

              <time title="Created: 2019-12-24 16:54:39" itemprop="dateCreated datePublished" datetime="2019-12-24T16:54:39+08:00">2019-12-24</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="fa fa-calendar-check-o"></i>
                </span>
                <span class="post-meta-item-text">Edited on</span>
                <time title="Modified: 2019-12-30 15:32:14" itemprop="dateModified" datetime="2019-12-30T15:32:14+08:00">2019-12-30</time>
              </span>
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="fa fa-folder-o"></i>
              </span>
              <span class="post-meta-item-text">In</span>
                <span itemprop="about" itemscope itemtype="http://schema.org/Thing">
                  <a href="/categories/SISR/" itemprop="url" rel="index">
                    <span itemprop="name">SISR</span>
                  </a>
                </span>
            </span>

          

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">

      
        <h1 id="Overview"><a href="#Overview" class="headerlink" title="Overview"></a>Overview</h1><p>　　SRCNN是深度用在超分辨率重建上的开山之作。其指出传统的基于稀疏矩阵的方法可以看做一个深层神经网络。SRCNN是一个轻量级的网络，只有三个卷积神经网络(CNN)。作者在训练时将图片转化为YCrCb三个通道，只对Y通道进行训练。因为人眼对Y通道比较敏感。</p>
<a id="more"></a>
<h1 id="Model"><a href="#Model" class="headerlink" title="Model"></a>Model</h1><img src="/images/SRCNN_model.png" class="[SRCNN模型示意图]" title="[14] [6] " alt="title text">
<p>整个SRCNN模型可以解释成三部分:</p>
<ul>
<li>Path extraction and Representation.提取图像Patch，进行卷积提取特征，类似于稀疏编码中的将图像patch映射到低分辨率字典中。</li>
<li>Non-linear mapping.将低分辨率的特征映射为高分辨率特征，类似于字典学习中的找到图像patch对应的高分辨字典。</li>
<li>Reconstruction.根据高分辨率特征进行图像重建。类似于字典学习中的根据高分辨率字典进行图像重建。</li>
</ul>
<p>　　首先用双立方插值对预先设计好的尺寸对一副低分辨率图像进行简单的画质提升处理，将处理后的图像表示为$Y$，原始图像为$X$。</p>
<h2 id="Path-extraction-and-Representation"><a href="#Path-extraction-and-Representation" class="headerlink" title="Path extraction and Representation"></a>Path extraction and Representation</h2><p>　　这一卷积层的作用可以表示为$\begin{equation}F_{1}\end{equation}$操作。</p>
<script type="math/tex; mode=display">\begin{equation}F_{1} = max(0, W_{1} * Y + B_{1})\end{equation}\label{eq1}</script><p>$W<em>{1}$是滤波器，$B</em>{1}$是bias. “*”是卷积操作。$W<em>{1}$对应一个$n</em>{1}$个滤波器大小是$c \times f<em>{1} \times f</em>{1}$, c是通道数，$f<em>{1}$是卷积核的大小。输出是一个$n</em>{1}$维的特征图。$B<em>{1}$是一个$n</em>{1}$维的向量。在经过每个滤波器之后添加ReLU($max(0, x)$)激活函数。</p>
<h2 id="Non-Linear-Mapping"><a href="#Non-Linear-Mapping" class="headerlink" title="Non-Linear Mapping"></a>Non-Linear Mapping</h2><p>　　第一层卷基层从图像补丁中提取$n<em>{1}$维特征。这一卷积层，从$n</em>{1}$维度非线性映射到$n<em>{2}$维度。这一卷积层的作用的可以表示为$F</em>{2}$算子。</p>
<script type="math/tex; mode=display">\begin{equation}F_{2} = max(0, W_{2} * F_{1}(Y) + B_{2})\end{equation}\label{eq2}</script><p>$W<em>{2}$对应一个$n</em>{2}$个滤波器大小是$n<em>{1} \times f</em>{2} \times f<em>{2}$, $n</em>{1}$是输入通道数，$f<em>{2}$是卷积核的大小。输出是一个$n</em>{2}$维的特征图。$B<em>{2}$是一个$n</em>{2}$维的向量。在经过每个滤波器之后添加ReLU($max(0, x)$)激活函数。</p>
<h2 id="Reconstruction"><a href="#Reconstruction" class="headerlink" title="Reconstruction"></a>Reconstruction</h2><p>　　这一重建卷基层的作用可以用$\begin{equation}F_{3}\end{equation}$算子。</p>
<script type="math/tex; mode=display">\begin{equation}F_{3} = max(0, W_{3} * F_{2}(Y) + B_{3})\end{equation}\label{eq3}</script><p>$W<em>{3}$对应一个c个滤波器大小是$n</em>{2} \times f<em>{3} \times f</em>{3}$, $n<em>{2}$是输入通道数，$f</em>{3}$是卷积核的大小。输出是一个c维的超分辨率重建图。$B_{3}$是一个c维的向量。</p>
<h1 id="Training-Detail"><a href="#Training-Detail" class="headerlink" title="Training Detail"></a>Training Detail</h1><p>　　端对端映射学习可以用$F$算子表示。网络参数用$\theta = {W<em>{1},W</em>{2},W_3,B_1,B_2,B_3}$表示。</p>
<h2 id="Loss-Function"><a href="#Loss-Function" class="headerlink" title="Loss Function"></a>Loss Function</h2><p>　　使用MSE(均方差误差)作为损失函数</p>
<script type="math/tex; mode=display">\begin{equation}L(\theta) = \frac{1}{n} \sum_{i=1}^{n} \parallel F(Y_i;\theta)-X_i \parallel ^2 \end{equation}\label{eq4}</script><p>n是batch_size的大小即是一批训练样本的大小。</p>
<h2 id="Learning-Rate"><a href="#Learning-Rate" class="headerlink" title="Learning Rate"></a>Learning Rate</h2><p>　　优化函数: 随机梯度下降（SGD）</p>
<script type="math/tex; mode=display">\begin{equation} \Delta_{i+1} = 0.9 \cdot \Delta_{i} + \eta \cdot \frac{\partial L}{\partial W_{i}^{l}}, W_{i+1}^{l} = W_{i}^{l} + \Delta_{i+1} \end{equation}\label{eq5}</script><p>$l$是层数 $l \in {1, 2, 3}$， i是迭代数, $\eta$是学习率。The filter weights of each layer are initialized by drawing randomly from a Gaussian distribution with zero mean and standard deviation 0.001 (and 0 for biases). 前两层的$lr=10^{-4}$ ,第三层的$lr=10^{-5}$.<strong>最后一层使用较小的学习率比较容易收敛</strong>。</p>
<h2 id="Training-Data"><a href="#Training-Data" class="headerlink" title="Training Data"></a>Training Data</h2><p>　　Basic network setting. $f_1=9, f_2=1, f_3=5, n_1=64, n_2=32$</p>
<p>　　Y channel pretrain. the other channel use the bicubic.</p>
<h1 id="PyTorch-Code"><a href="#PyTorch-Code" class="headerlink" title="PyTorch Code"></a>PyTorch Code</h1><p>　　下面是PyTorch实现的代码。<br><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> torch.nn <span class="keyword">as</span> nn</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">SRCNN</span><span class="params">(nn.Module)</span>:</span></span><br><span class="line">    <span class="function"><span class="keyword">def</span> <span class="title">__init__</span><span class="params">(self)</span>:</span></span><br><span class="line">        super(SRCNN,self).__init__()</span><br><span class="line">        <span class="comment"># here not like the paper, I use the padding</span></span><br><span class="line">        <span class="comment"># here the weight init is not implement</span></span><br><span class="line">        self.conv1 = nn.Conv2d(<span class="number">1</span>,<span class="number">64</span>,kernel_size=<span class="number">9</span>,padding=<span class="number">4</span>);</span><br><span class="line">        self.relu1 = nn.ReLU();</span><br><span class="line">        self.conv2 = nn.Conv2d(<span class="number">64</span>,<span class="number">32</span>,kernel_size=<span class="number">1</span>,padding=<span class="number">0</span>);</span><br><span class="line">        self.relu2 = nn.ReLU();</span><br><span class="line">        self.conv3 = nn.Conv2d(<span class="number">32</span>,<span class="number">1</span>,kernel_size=<span class="number">5</span>,padding=<span class="number">2</span>);</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">def</span> <span class="title">forward</span><span class="params">(self,x)</span>:</span></span><br><span class="line">        out = self.conv1(x)</span><br><span class="line">        out = self.relu1(out)</span><br><span class="line">        out = self.conv2(out)</span><br><span class="line">        out = self.relu2(out)</span><br><span class="line">        out = self.conv3(out)</span><br><span class="line">        <span class="keyword">return</span> out</span><br><span class="line"></span><br><span class="line"></span><br></pre></td></tr></table></figure></p>
<h1 id="Reference"><a href="#Reference" class="headerlink" title="Reference"></a>Reference</h1><p><a href="http://arxiv.org/abs/1501.00092" target="_blank" rel="noopener">论文地址</a><br><a href="https://github.com/basher667/pytorch_srcnn" target="_blank" rel="noopener">参考代码</a></p>

    </div>

    
    
    

      <footer class="post-footer">

        


        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/2019/10/19/e6-9c-80-e8-bf-91-e7-82-b9-e5-af-b9-e9-97-ae-e9-a2-98/" rel="prev" title="最近点对问题">
      <i class="fa fa-chevron-left"></i> 最近点对问题
    </a></div>
      <div class="post-nav-item">
    <a href="/2019/12/27/GFRNet/" rel="next" title="GFRNet">
      GFRNet <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
  

  </div>


          </div>
          

<script>
  window.addEventListener('tabs:register', () => {
    let activeClass = CONFIG.comments.activeClass;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>

        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          Table of Contents
        </li>
        <li class="sidebar-nav-overview">
          Overview
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#Overview"><span class="nav-number">1.</span> <span class="nav-text">Overview</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#Model"><span class="nav-number">2.</span> <span class="nav-text">Model</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#Path-extraction-and-Representation"><span class="nav-number">2.1.</span> <span class="nav-text">Path extraction and Representation</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Non-Linear-Mapping"><span class="nav-number">2.2.</span> <span class="nav-text">Non-Linear Mapping</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Reconstruction"><span class="nav-number">2.3.</span> <span class="nav-text">Reconstruction</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="#Training-Detail"><span class="nav-number">3.</span> <span class="nav-text">Training Detail</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#Loss-Function"><span class="nav-number">3.1.</span> <span class="nav-text">Loss Function</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Learning-Rate"><span class="nav-number">3.2.</span> <span class="nav-text">Learning Rate</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Training-Data"><span class="nav-number">3.3.</span> <span class="nav-text">Training Data</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="#PyTorch-Code"><span class="nav-number">4.</span> <span class="nav-text">PyTorch Code</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#Reference"><span class="nav-number">5.</span> <span class="nav-text">Reference</span></a></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
  <p class="site-author-name" itemprop="name">Z.J. Jiang</p>
  <div class="site-description" itemprop="description">about the single image super-resolution and face hallucination</div>
</div>
<div class="site-state-wrap motion-element">
  <nav class="site-state">
      <div class="site-state-item site-state-posts">
          <a href="/archives/">
        
          <span class="site-state-item-count">42</span>
          <span class="site-state-item-name">posts</span>
        </a>
      </div>
      <div class="site-state-item site-state-categories">
            <a href="/categories/">
          
        <span class="site-state-item-count">11</span>
        <span class="site-state-item-name">categories</span></a>
      </div>
      <div class="site-state-item site-state-tags">
            <a href="/tags/">
          
        <span class="site-state-item-count">1</span>
        <span class="site-state-item-name">tags</span></a>
      </div>
  </nav>
</div>
  <div class="links-of-author motion-element">
      <span class="links-of-author-item">
        <a href="https://github.com/jzijin" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;jzijin" rel="noopener" target="_blank"><i class="fa fa-fw fa-github"></i>GitHub</a>
      </span>
      <span class="links-of-author-item">
        <a href="/atom.xml" title="RSS → &#x2F;atom.xml"><i class="fa fa-fw fa-rss"></i>RSS</a>
      </span>
  </div>


  <div class="links-of-blogroll motion-element">
    <div class="links-of-blogroll-title">
      <i class="fa fa-fw fa-link"></i>
      Links
    </div>
    <ul class="links-of-blogroll-list">
        <li class="links-of-blogroll-item">
          <a href="http://www.njust.edu.cn/" title="http:&#x2F;&#x2F;www.njust.edu.cn" rel="noopener" target="_blank">南京理工大学</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="http://ehall.njust.edu.cn/new/index.html" title="http:&#x2F;&#x2F;ehall.njust.edu.cn&#x2F;new&#x2F;index.html" rel="noopener" target="_blank">南京理工大学智慧服务</a>
        </li>
        <li class="links-of-blogroll-item">
          <a href="http://lib.njust.edu.cn/" title="http:&#x2F;&#x2F;lib.njust.edu.cn&#x2F;" rel="noopener" target="_blank">南京理工大学图书馆</a>
        </li>
    </ul>
  </div>

      </div>

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        

<div class="copyright">
  
  &copy; 
  <span itemprop="copyrightYear">2022</span>
  <span class="with-love">
    <i class="fa fa-user"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">Z.J. Jiang</span>
</div>
  <div class="powered-by">Powered by <a href="https://hexo.io/" class="theme-link" rel="noopener" target="_blank">Hexo</a> v4.2.0
  </div>
  <span class="post-meta-divider">|</span>
  <div class="theme-info">Theme – <a href="https://pisces.theme-next.org/" class="theme-link" rel="noopener" target="_blank">NexT.Pisces</a> v7.6.0
  </div>

        








      </div>
    </footer>
  </div>

  
  <script src="/lib/anime.min.js"></script>
  <script src="/lib/velocity/velocity.min.js"></script>
  <script src="/lib/velocity/velocity.ui.min.js"></script>

<script src="/js/utils.js"></script>

<script src="/js/motion.js"></script>


<script src="/js/schemes/pisces.js"></script>


<script src="/js/next-boot.js"></script>




  















  

  
      
<script type="text/x-mathjax-config">
    MathJax.Ajax.config.path['mhchem'] = '//cdn.jsdelivr.net/npm/mathjax-mhchem@3';

  MathJax.Hub.Config({
    tex2jax: {
      inlineMath: [ ['$', '$'], ['\\(', '\\)'] ],
      processEscapes: true,
      skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code']
    },
    TeX: {
        extensions: ['[mhchem]/mhchem.js'],
      equationNumbers: {
        autoNumber: 'AMS'
      }
    }
  });

  MathJax.Hub.Register.StartupHook('TeX Jax Ready', function() {
    MathJax.InputJax.TeX.prefilterHooks.Add(function(data) {
      if (data.display) {
        var next = data.script.nextSibling;
        while (next && next.nodeName.toLowerCase() === '#text') {
          next = next.nextSibling;
        }
        if (next && next.nodeName.toLowerCase() === 'br') {
          next.parentNode.removeChild(next);
        }
      }
    });
  });

  MathJax.Hub.Queue(function() {
    var all = MathJax.Hub.getAllJax(), i;
    for (i = 0; i < all.length; i += 1) {
      element = document.getElementById(all[i].inputID + '-Frame').parentNode;
      if (element.nodeName.toLowerCase() == 'li') {
        element = element.parentNode;
      }
      element.classList.add('has-jax');
    }
  });
</script>
<script>
  NexT.utils.getScript('//cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML', () => {
    MathJax.Hub.Typeset();
  }, window.MathJax);
</script>

    

  

</body>
</html>
